We study carbon nanotubes (CNTs) connected to Nb superconducting electrodes through a thin (less than 5 nm) Pd layer. We show that the carbon nanotubes form nanoscale point contacts detecting superconducting proximity effect via Andreev reflection, where an electron injected from the nanotube is reflected as a hole at the Pd/Nb interface and a Cooper pair forms in the Nb. Our data cannot be quantitatively explained by a single interface model, where only one interface between the CNT and a superconducting electrode is considered. Instead, we present a quantitative analysis that includes two separate interfaces-CNT/Pd and Nb/Pd-at each end of the CNT. The data can be used to determine the transparencies of the Pd/Nb and CNT/Pd interfaces.